Unit root problem, solution of difference equations Simple deterministic model, question of unit root

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1 Unit root problem, solution of difference equations Simple deterministic model, question of unit root (1 φ 1 L)X t = µ, Solution X t φ 1 X t 1 = µ X t = A + Bz t with unknown z and unknown A (clearly X t 1 = A + Bz t 1 ) 0 = X t φ 1 X t 1 µ = A + Bz t φ 1 (A + Bz t 1 ) µ = A + Bz t φ 1 ( A + B z zt ) µ ( = µ + A(1 φ 1 ) +B 1 φ ) 1 }{{} z =0 }{{} z t

2 z = φ 1, A = µ 1 φ 1 Elimination B by the initial condition X t0 = µ 1 φ 1 + Bφ t 0 1 B = X t = µ + 1 φ 1 ( X t µ ) 1 φ 1 X t = µ 1 φ 1 + Bφ t 1 [ X t0 µ 1 φ 1 ] [ X t0 µ = φ t φ 1 ( X t0 µ 1 φ 1 X t µ 1 φ 1 X t0 µ = φ t t φ 1 ] φ t 0 1 φ t 1 ) φ t t 0 1 Qualitative difference for φ 1 < 1 and φ 1 > 1 (unit root means z = φ 1 = 1) φ 1 = 0.8: t 1 t 0 = 0, φ 0 1 = 1; t t 0 = 1, φ 1 1 = 0.8; t t 0 = 2, φ 2 1 = 0.64; t t 0 = 3; φ 3 1 φ 1 = 1.2: t t 0 = 10, φ 10 1 = 0.52; φ101 = 6.19; φ100 = 0.107, φ1001 = ; 1 = ;

3 (1 φ 1 (1/z) φ 2 (1/z) 2 φ 3 (1/z) 3... φ p (1/z) p )z t X 0 = 0 Unit root signature of nonstationary behaviour z = φ 1 = 1 Take homogeneous equation (1 φ 1 L)X t = 0, (1 L)X t = 0 formally via the "trick" L = 1/z we get characteristic equation (1 φ 1 /z) = 0 where the unit root is achieved for z = φ 1 = 1 Generalization for AR(p) without stochastic component: p X t φ i X t i = 0 i=1 X t φ 1 X t 1 φ 2 X t 2 φ 3 X t 3... φ p X t p = 0 (1 φ 1 L φ 2 L 2 φ 3 L 3... φ p L p )X t = 0 expected form of solution X t = z t X 0, t = 0, 1, 2...

4 [ 1 φ 1 (1/z) φ 2 (1/z) 2 φ 3 (1/z) 3... φ p (1/z) p] z t X 0 = 0 characteristic equation z p φ 1 z p 1 φ 2 z p 2 φ 3 z p 3... φ p = 0 generally has exactly p roots (z λ 1 )(z λ 2 )... (z λ p ) = 0 p (z λ k ) k = 0 k=1 For non-identical (also complex) roots X t = B 1 λ t 1 + B 2 λ t B p λ t p when roots are identical for e.g. p = 2 X t = B 1 λ t 1 + B 2 tλ t 1

5 Augmented Dickey-Fuller (ADF) unit root test (Rozšírený DF test) The test is widely used by practioners. In statistics, the traditional DickeyFuller test tests (1979) whether a unit root is present in an autoregressive model. In time series models in econometrics (the application of statistical methods to economics), a unit root is a feature of processes that evolve through time that can cause problems in statistical inference if it is not adequately dealt with. Such a process is non-stationary. (If the other roots of the characteristic equation lie inside the unit circle that is, have a modulus (absolute value) less than one then the first difference of the process will be stationary). Other words: series of the first difference (previously considered integrated of order one) turned to be stationary.

6 R: test available in adf.test() tseries() [package for Time series analysis and computational finance] The testing procedure is applied to the model structure X t = α }{{} drift + δ t }{{} lin.trend + ϱ }{{} X t 1 + par. tested p ψ j X t j X t j = X t j X t j 1, X t = X t X t 1 number of lags of autoregression; default: p = (length of d.s. 1) 1/3 ; (example: length of data series = 65, p = 4); Swert criterion: 12 (length of d.s./100) 1/4 Alternative form: p X t = α + δt + (1 + ϱ) X }{{} t 1 + ψ j X t j φ 1 in former notation j=1 p j=1 ψ j X t j the autoregression terms absorbing dynamic j=1

7 H 0 : ϱ = 0 for given model has a unit root, process is I (1), i.e. the data needs to be differenced to make it stationary; H 1 : ϱ < 1 i.e. the data is trend stationary, process is I (0) and needs to be analyzed by means of using a trend in the regression model instead of differencing the data; The unit root test is carried out under H 0 : ϱ = 0 against the alternative hypothesis of ϱ < 0. Note 1: Caution: Sometimes if data is exponentially trending then you might need to take the log of the data first before differencing it. Note 2: ϱ = 0 means X t X t 1 (i.e. random walk process)

8 test statistic [one-sided left tail test]: ADF τ = ˆϱ SE(ˆϱ) (remind that t score = ˆϱ SE(ˆϱ) ) SE(ˆϱ) is the usual standard error estimate critical value: t 1 α(n) If the test statistic is less than the critical value t 1 α(n), then the H 0 ( ϱ = 0) is rejected and no unit root is present. Example: A model that includes a constant and a time trend is estimated using sample of n = 50 observations and yields the ADF τ statistic of This is more negative than the tabulated t = 3.50, so at the 95 per cent level the null hypothesis of a unit root will be rejected.

9 R implementation > x=arima.sim(n = 1000, list(ma = c(-0.2, 0.2)),sd=1) > adf.test(x) Augmented Dickey-Fuller Test data: x Dickey-Fuller = , Lag order = 9, p-value = 0.01 alternative hypothesis: stationary Warning message: In adf.test(x) : p-value smaller than printed p-value [Note: rejection of null hypothesis that data has a unit root]

10 > x= arima.sim(n = 1000, list(ma = c(-0.2, 0.2)),sd=1) + 0.1*(c(1:1000)^3) > adf.test(x) Augmented Dickey-Fuller Test data: x Dickey-Fuller = , Lag order = 9, p-value = 0.99 alternative hypothesis: stationary Warning message: In adf.test(x) : p-value greater than printed p-value

11 KPSS test Kwiatkowski, Phillips, Schmidt, Shin (1992) observed time series x t is expressed as the sum x t = T t + RW t + u t : - deterministic trend T t - pure random walk RW t (i.e. I (1)), RW t = RW t 1 + ɛ t - stationary error term u t (i.e. I (0)) H 0 : σ 2 ɛ = 0 observable time series is stationary around a deterministic trend, x t is I (0) H 1 : σ 2 ɛ > 0

12 in R: > s<-rnorm(1000) > kpss.test(s) > > KPSS Test for Level Stationarity > > data: s > KPSS Level = , Truncation lag parameter = 7, > p-value = stationarity cannot be rejected here...

13 Second order Stochastic Difference Equation and Business Cycles Samuelson (1939) applied only second order difference equations to explain the business cycles: Investment I t is dependent on the changes in income rather then the level of income I t = b 1 (Y t 1 Y t 2 ) Consumption C t depends on the past income by the first order nonhomogeneous equation C t = b 2 Y t 1 + b 3 Production/consumption balance with error term Y t = C t + I t + ɛ t

14 Consequence - accelerator principle Y t = b 1 (Y t 1 Y t 2 ) + b 2 Y t 1 + b 3 + ɛ t simulation in R b3<-1.04 b2<-0.2 b1<-1.01 Y1<-1.0 Y2<-1.04 Y0<-1 for(i in 1:200){ Y2<-Y1 Y1<-Y0 Y0<-(b1*(Y1-Y2)+b2*Y1+b3) print(paste(y0)) }

15 Y "./Simul_Samuelson0.txt" time b3<-1.04 b2<-0.2 b1<-1.01 Y1<-1.0 Y2<-1.04 Y0<-1 for(i in 1:200){ Y2<-Y1 Y1<-Y0 Y0<-(b1*(Y1-Y2)+b2*Y1+b3) + rnorm(1,mean=0,sd=0.3) }

16 8 6 "./Simul_Samuelson1.txt" "./Simul_Samuelson2.txt" 4 Y time

17 Solution of the homogeneous (non-stochastic) Samuelson s system z =..., b 3 = 0 Y t = z t Y 0 Y t = b 1 (Y t 1 Y t 2 ) + b 2 Y t 1 z t Y 0 = b 1 (z t 1 Y 0 z t 2 Y 0 ) + b 2 Y 0 z t 1 ( 1 1 = b 1 z 1 ) 1 z 2 + b 2 z lag operator formally produces 1/z ( 1 b 1 z 1 ) 1 z 2 + b 2 z 1 = 0 b 1 + b 2 b = 0

18 z 2 (b 1 + b 2 )z + b 1 = 0 z 1,2 = b 1 + b 2 ± (b 1 + b 2 ) 2 4b 1 2 b 1 = 1.01, b 2 = 0.2, b 3 = 1.04 z 1,2 = 1.21 ± = ± i z 1,2 = , arccos (0.605/ z 1,2 ) = (2π) Y t = Y 0 z t = ( ) t [cos(±(2π)0.147t) + i sin(±(2π)0.147t)] ( = ( ) t exp(i2π0.147t) = ( ) t exp i2π t ) 6.8

19 X Qualitative check of the period x x x ACF Lag

20 Comparison Samuelson s accelerator (b 3 = 0) with ARIMA(p,d,.) (1 φ 1 φ 2 L)(1 L)Y t = (1 L Lφ 1 + L 2 φ 1 L 2 φ 2 + L 3 φ 2 ) Y t = b 1 (Y t 1 Y t 2 ) + b 2 Y t 1 Try p = 1, d = 1 ( ) p 1 φ i L i (1 L) d Y t = 0 i=1 (1 φ 1 L)(1 L)Y t = (1 φ 1 L)(Y t Y t 1 ) = Y t Y t 1 φ 1 (LY t LY t 1 ) Try p = 2, d = 1 = Y t Y t 1 φ 1 (Y t 1 Y t 2 )

21 = Y t Y t 1 φ 1 Y t 1 + φ 1 Y t 2 φ 2 Y t 2 + φ 2 Y t 3 ) Try p = 2, d = 0 (1 φ 1 L φ 2 L 2 )Y t = Y t φ 1 Y t 1 φ 2 Y t 2 Y t φ 1 Y t 1 φ 2 Y t 2 = 0 Y t = φ 1 Y t 1 + φ 2 Y t 2 Y t = b 1 (Y t 1 Y t 2 ) + b 2 Y t 1 = (b 1 + b 2 ) Y }{{} t 1 + ( b 1 ) }{{} φ 1 Conclusion: Model is transformable to ARIMA(2,0,q) i.e. d = 0 model φ 2 Y t 2

22 Box Jenkins metodology 1. identification of model, mostly AR(1) making sure that model is stationary using plots of ACF, and PACF 2. estimation of the model parameters: say x t = 0.68x t 1 + ɛ t, ˆσ = using the maximum likelihood non-linear least-squares 3. diagnostics of the model: prediction abilities, Ljung-Box test Model identification by means of autocorrelation function: U-shaped after first q AR(p) MA(q) ARMA(p, q) ρ k k 0 does not exist k 0 = q k 0 does not exist ACF ρ k is U shaped ρ k becomes ρ kk k 0 = p k 0 does not exist k 0 does not exist PACF ρ kk is limited ρ kk becomes by U-shaped curve U-shaped after the first

23 Table: U denotes the curve of geometrically or sin-like decaying amplitudes. Rule 1 If the series has positive autocorrelations out to a high number of lags, then it probably needs a higher order of differencing. Rule 2 If the lag-1 autocorrelation is zero or negative, or the autocorrelations are all small a patternless, then the series does not need a higher order of differencing. If the lag-1 autocorrelation is -0.5 or more negative, the series may be overdifferenced. Beware of overdifferencing. Rule 3 The optimal order of differencing is then often the order at which the standard deviation is lowest. Rule 4 A model with no orders of differencing assumes that the original series is stationary. Rule 5 A model with no orders of differencing normally includes a constant term Rule 6 If the part of PACF of the differenciated series displays a sharp cutoff and/or the lag 1 autocorrelation is positive, i.e. the

24 series appears slightly underdifferenced then consider adding one or more AR terms to the model Rule 7 The autocorrelation function ACF of the differenced series displays a sharp cutoff and or the lag-1 autocorrelation function is negative. Rule 8 It is possible for an AR term and MA term to cancel each other s effects, so if a mixed AR-MA model seems to fit data also try a model with one fewer AR term. Rule 9 If there is a unit root in the AR part of the model the sum of AR coefficients is almost exactly 1 Rule 10 If the long-term forecasts appear erratic or unstable, there may be a unit root. Rule 11 If the series has a strong and consistent seasonal pattern, then you should use an order of seasonal differencing.

25 Autoregression of simulated time series of the price formation process Model of random process: p 0, p 1, p 2, p 3,..., p t,... if p t > p fun { p t+1 = p t 0.99} else p t+1 = p t if p t+1 < p fun {p t+1 = p t } else p t+1 = p t+1 p t+1 = p t+1 exp [ N(0, 0.01) ]

26 R project: simulation of price dynamics pfun<-1 # "fundamental" price p<-(pfun+0.01) # initial price setting vp<-p # auxilliary initial vector setting for pl Ns<-1000 # number of steps time=c(0:ns) # vector of times for(i in 1:Ns){ if(p>pfun) p<-(p * 0.99) if(p<pfun) p<-(p * 1.01) # deterministic reduction # deterministic increase p<-(p*exp(rnorm(1,mean=0,sd=0.01))) #randomization vp<-cbind(vp,p) } vp<-as.vector(vp)

27 pdf("tp_simul.pdf") plot(time,vp, type="l", pch=21, xlab="time", ylab="p") dev.off() p time

28 Testing of simulated price series for stationarity adf.test(vp) Augmented Dickey-Fuller Test data: vp Dickey-Fuller = , Lag order = 9, p-value = 0.01 alternative hypothesis: stationary Warning message: In adf.test(vp) : p-value smaller than printed p-value conclusion: reject unit root

29 Selection of the "best" arima model based on the Aikake information criterion The Akaike information criterion is a measure of the relative goodness of fit of a statistical model. It is grounded in the concept of information entropy, in effect offering a relative measure of the information lost when a given model is used to describe reality. In the general case, the AIC is AIC = 2k 2 ln(l), where k is the number of parameters in the statistical model, and L is the maximized value of the likelihood function for the estimated model. To apply AIC in practice, we start with a set of candidate models, and then find the models corresponding AIC values. There will almost always be information lost due to using one of the candidate models to represent the "true" model. We wish to select, from among n candidate models, the model that minimizes the estimated information loss. Denote the AIC values of the candidate models by AIC 1, AIC 2, AIC 3,..., AIC n. Let AIC min be the minimum of those values. Then e (AIC min AIC i )/2 can be interpreted as the relative probability that the i-th model minimizes the (estimated) information loss.

30 ########################################################################## Fit an ARIMA model to a univariate time series. Usage arima(x, order = c(0, 0, 0), seasonal = list(order = c(0, 0, 0), period = NA),... optim.control = list(), kappa = 1e6) # Example: R code for selection of the "best" model # Aikake s information criterion, select among 10 candidate models a1=arima(vp,order=c(0,0,0)) a2=arima(vp,order=c(0,0,1)) a3=arima(vp,order=c(0,1,0)) a4=arima(vp,order=c(1,0,0)) a5=arima(vp,order=c(1,1,0)) a6=arima(vp,order=c(0,1,1)) a7=arima(vp,order=c(1,0,1)) a8=arima(vp,order=c(1,1,1)) a8=arima(vp,order=c(0,0,2)) a9=arima(vp,order=c(0,2,0)) a10=arima(vp,order=c(2,0,0)) veca=as.vector(cbind(a1$aic,a2$aic,a3$aic,a4$aic, a5$aic,a6$aic,a7$aic,a8$aic,a9$aic,a10$aic)) which.min(veca)

31 Model Outputs: > which.min(veca) [1] 4 Check the best: > a4 Call: arima(x = vp, order = c(1, 0, 0)) Coefficients: ar1 intercept s.e sigma^2 estimated as : log likelihood = , aic = !!! intercept indicates fundamental price p_fun=1

32 ### Check another one = not the best > a3 Call: arima(x = vp, order = c(0, 1, 0)) sigma^2 estimated as : log likelihood = , aic =

33 Forecast by Arima predict.arima {stats} R Documentation Forecast from ARIMA fits Description: Forecast from models fitted by arima. predict(object, n.ahead = 1, newxreg = NULL, se.fit = TRUE,...) Arguments: object The result of an arima fit. n.ahead The number of steps ahead for which prediction is required.

34 Make the prediction: > a4=arima(x=vp, order=c(1,0,0)) > predict(a4,n.ahead=6) \$pred Time Series: Start = 1002 End = 1007 Frequency = 1 [1] \$se Time Series: Start = 1002 End = 1007 Frequency = 1 [1]

35 Testing of forecast vp1=vp[1:990] > a4=arima(vp1,order=c(1,0,0)) > vp2=vp[991:1000] > pre=predict(a4,n.ahead=10) Time Series: Start = 991 End = 1000 Frequency = 1 [1] [9] \$se Time Series: Start = 991 End = 1000 Frequency = 1 [1] [7] > vp2 [1] [8]

36 vpred=pre\$pred[1:10] > vp2-vpred [1] [6] compare errors with a mean standard deviation (vp2-vpred)/mean(se1) [1] [7]